Probability

[Heat]

1. Suppose you're on a game show and you're given the choice of three doors. Behind one is a car, behind each of the others is a goat. You pick a door, say door A, and the host, who knows what's behind the other doors, opens another door, say B, which has a goat. He then says : "Do you want to switch to door C?" Is it to your advantage to take the switch?'

2. There is a card in a hat. It is either the ace of spades or the king of spades, with equal probability. You take another identical ace of spades and throw it into the hat. You then choose a card at random from the hat. You see it is an ace. What are the odds the original card in the hat was an ace?

3. You meet a woman who has a son in your child's class. She says "I have two children". So, at this point, you know that the woman has two children and at least one of those two children is a boy. What is the probability that this woman has two boys?

4. The same woman's two children come to visit you. The boy from your child's class enters your house and the woman's second child knocks on the door. What is the probability that the woman has two boys?

5. There's a lottery with 10,000 "scratch-off-the-dot" tickets with one ticket being the winner of a car, the other 9,999 tickets being losers.

Ten thousand people buy the tickets, including you. 9,998 scratch off the dots on their tickets and find the message YOU LOSE. Should you offer big money to the remaining ticketholder to exchange tickets with you?

Spoilerize any answers...

Question 1:

Spoiler:

Yes, switch. There is a 2/3 chance that Door C will be the winner.

Question 2:

Spoiler:

The answer is 2/3.

Question 3:

Spoiler:

The answer is 1/3.

Question 4:

Spoiler:

The answer is 1/2.

Question 5:

Spoiler:

5. No (you both have the same probability of winning).

[Minor Threat]

I'd whip out my STOAP infused gum and vigorously chew until I had the answer.....

[Charlie Goose]

1. There are 100 threads created each day in Other. What is the probability that this one will suck?

Question 1:

Spoiler:

100%

[GatorDeb]

Spoiler:

Why are three and four not the same?

[4KRG]

Please show your math, otherwise it looks like you cheated

[Bushdog]

Originally Posted by GatorDeb

Spoiler:

Why are three and four not the same?

Spoiler:

Yeah, this one kind of depends on the perspective.

The possible combinations of two kids are:
1) MM
2) MF
3) FM
4) FF

In three, we can exclude possibility #4 (FF) thus 1/3rd.

In 4, we are taking only the gender of this child into account, we are essentially assuming the gender of the first child as male. By doing this, we eliminate choice 4 still (FF) as well as choice 3 (FM, the first child being a female since the first child is a male based on observation)).

In three, the first child could be of either gender, thus options 2 and 3 are both valid. Not so in the 2nd scenario.

[Heat]

Show my math on which one?

Questions 1 and 3

Spoiler:

are classic Marilyn Von Savant questions (from the early to mid 1990s), both of these stumped quite a few people including math professors but the answers I showed are correct.

The key to question 1 is that the game show host knows which door is not a winner. The key to question 3 is that you don’t know whether the boy is the older or younger child thus the possibilities are still boy-girl / girl-boy / boy-boy. There’s a one-third chance that it’s boy-boy. (see Bushdog's explanation).

Question 2

Spoiler:

is simple odds. There are only three possibilities:
a. You put an ace in the hat and pull this ace out.
b. You put an ace in the hat and pull the 2nd ace out.
c. You put a king in the hat and pull the 2nd ace out.
There’s a 2/3 chance that the original card was an ace.

Question 4:

Spoiler:

In question 3 you didn’t know whether the child was the older or younger child. In question 4 you’re looking at only one child, the second child, so it’s back to a 50/50 chance.

Question 5:

Spoiler:

I don’t think that anybody had any trouble with Question 5.

[BravesMG]

It's been a while since my last probabilities class, but #1 confuses me.

Spoiler:

Isn't there a principle that if all elements of an equation are unknown, their properties are equal no matter what else in the equation changes? I remember a question like this on an exam, and I'm pretty sure I got it wrong then too, but if it was true that your odds do increase by switching from your initial selection to which ever door you did not choose, wouldn't it be the same if you reversed your initial selections? So it couldn't possibly increase your odds in both cases? I know I'm probably wrong on this, but I could use some help.

I just read the above post regarding #1, but I'm still confused as to how

Spoiler:

because the host knows what is behind each door, why that matters to the probability of the contestant.

[Bushdog]

This is a long discussion and I'm too lazy to get into it.

Google "Monty Hall Problem" and all your questions will be answered.

[das Monkey]

#1 is one of my favorites. Whenever someone is posturing about how smart he thinks he is, I'll break it out. A guy at work wrote a computer program to "prove me wrong". That didn't work out the way he wanted.

bravesmg, the question is worded somewhat poorly above. The key to the problem is that the host always shows you a bad prize (goat) before asking you to switch.

das

[Mopower]

Spoiler:

I just wanted to spoilerize something like everyone else has.

[das Monkey]

If you want an explanation for #1 that's simple to understand:

Spoiler:

You have 3 doors. You pick 1. The probability you chose the car is 1/3. The probability you didn't is 2/3. Whether he shows you what's behind one of the doors or all the doors for that matter, those numbers don't change. There's a 2/3 chance you picked the wrong door. By eliminating one of the doors you didn't pick and allowing you to switch to the other, he's essentially allowing you to pick both of those doors at the same time.

das

[kvrdave]

Originally Posted by bravesmg

I just read the above post regarding #1, but I'm still confused as to how

Spoiler:

because the host knows what is behind each door, why that matters to the probability of the contestant.

Here is the dumbed down version that I use because I am dumbed down....

Spoiler:

When you initially start, you have a 33% chance of getting the correct door. Once a door is taken away, you can either sit on your original 33% chance of being right or move to a 50% chance of being right. Yeah, I know, it seems like word trickery, but it ain't....it's powerful magic.

[BravesMG]

I know it was poorly worded, I'm usually right on probability, I've got an Economics degree, but this always has baffled me. I'm going into the web explanations now.

[Kittydreamer]

Originally Posted by Mopower

Spoiler:

I just wanted to spoilerize something like everyone else has.

Spoiler:

I don't know how to do spoiler tags and had to quote you so I could figure it out. I also don't know how to do probablity stuff. I'm useless.

[deadlax]

Spoiler:

#1 is hands down the dumbest "brain teaser" of all-time. Granted you get to "choose" both doors, after door B is open you have a 50/50 shot at guessing the correct door. It is either a or c. Once that information has been given to all participants in the game the odds have been shifted.

You can try to stump brilliant people all the time with this. And I get why it makes "statistical" sense to make the switch. But it doesn't make common sense to make the switch.

[das Monkey]

• deadlax •

Spoiler:

#1 is hands down the dumbest "brain teaser" of all-time. Granted you get to "choose" both doors, after door B is open you have a 50/50 shot at guessing the correct door. It is either a or c. Once that information has been given to all participants in the game the odds have been shifted.

You can try to stump brilliant people all the time with this. And I get why it makes "statistical" sense to make the switch. But it doesn't make common sense to make the switch.

Too many people think with their brains instead of feeling with their gut.

das

[Bushdog]

Spoiler:

Probabilistic sense, not statistical sense. They are different things.

[OldDude]

Spoiler:

I don't buy the answer on number 1. The logic (not the numvers) of #1 and #5 are identical. The original odds are changed once you rule out a bunch of losers, but the new odds are equal amongst the remaining choices as no winners have been revealed. Both the remaining doors have 50% odds.
Actually, my analysis assumes the host ALWAYS shows you a goat, and offers to let you switch. If he does so selectively, then the game is rigged. The odds of whether it is rigged for you or against can not be estimated from the problem statement.

Further, the two kids problems, #3 and #4 have the same logic and same answer 50%. Nothing about the boy you know about changes the odds of the other kid. Actually the odds of a boy are slightly higher than 50% based on birth rates. I'm too lazy to go dig for the exact number.

[Heat]

All-right, I’ll make the call. Once page 2 starts, spoiler tags aren’t needed anymore.

Originally posted by Old Dude

Spoiler:

Further, the two kids problems, #3 and #4 have the same logic and same answer 50%. Nothing about the boy you know about changes the odds of the other kid. Actually the odds of a boy are slightly higher than 50% based on birth rates. I'm too lazy to go dig for the exact number.

It's something like 109 boys for every 100 girls, but girls have a higher childhood survival rate.

Spoiler:

And it's tricky, but the key is that you don't know whether the boy in the classroom is the older or younger sibling. When the kid knocks on the door, order doesn't matter since you are only looking at one child at that point.

[talemyn]

Originally Posted by OldDude

Spoiler:

I don't buy the answer on number 1. The logic (not the numvers) of #1 and #5 are identical. The original odds are changed once you rule out a bunch of losers, but the new odds are equal amongst the remaining choices as no winners have been revealed. Both the remaining doors have 50% odds.
Actually, my analysis assumes the host ALWAYS shows you a goat, and offers to let you switch. If he does so selectively, then the game is rigged. The odds of whether it is rigged for you or against can not be estimated from the problem statement.

Further, the two kids problems, #3 and #4 have the same logic and same answer 50%. Nothing about the boy you know about changes the odds of the other kid. Actually the odds of a boy are slightly higher than 50% based on birth rates. I'm too lazy to go dig for the exact number.

I agree with you on the first question. I always hated that one (and argued with my professor about it for a long time). I still don't agree with the "official" answer.

Of course I was hesitant to post that I agreed with you because you are completely wrong about questions 3 and 4.

[OldDude]

Originally Posted by Heat

[b]

Spoiler:

And it's tricky, but the key is that you don't know whether the boy in the classroom is the older or younger sibling. When the kid knocks on the door, order doesn't matter since you are only looking at one child at that point.

Spoiler:

I don't see that the order matters. You know one child is a boy your child's age (close enough for same class) and you don't know shit about the other child, so the odds are (roughly 50/50.

Are these questions from this month's "Lies, Damned lies, Statistics."

Edit: Ok, I'm wrong on #1, and the Wikipedia entry offers analysis from several points of view. At least one will be compelling.

[shaun3000]

I agree that #1 is stupid.

[El Scorcho]

As always, wikipedia explains things better than you shlubs.

[dork]

Originally Posted by OldDude

Spoiler:

I don't see that the order matters. You know one child is a boy your child's age (close enough for same class) and you don't know shit about the other child, so the odds are (roughly 50/50.

#3 is poorly phrased, but the point is that

Spoiler:

you supposedly don't know which of her two sons is in your child's class (if she, in fact, has two sons). So that adds another possible event. However, in #4 you already see one son so the probability that the other child is a boy is now 1/2. It's like you collapsed the wave function.